II Nonlinear wave equations 2.1 Introduction

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II Nonlinear wave equations

2.1 Introduction

• Introduction • Solitary waves

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Introduction

Linear wave equations

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Introduction

Linear wave equations

• Simplest Linear

u_t – cu_x = 0 or u_t + cu_x = 0 u(x,t) = f(x+ct) or u(x,t) = f(x-ct) • Simplest Dispersive, Dissipationless

u_t + cu_x + au_xxx = 0

u(x,t) = exp[i(kx – ωt)] ω = ck - ak3

• Simplest Nondispersive, Dissipative u_t + cu_x - au_xx = 0

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Introduction

Nonlinear wave equations

• Simplest Nonlinear u_t + (1+u)u_x = 0

u(x,t) = f(x-(1+u)t)

Sharpens at leading and trailing edges (shock formation) • Korteweg deVries (KdV) Equation (1895)

u_t + (1+u)u_x + u_xxx = 0

Solitary wave/soliton behaviour

Dispersion and tendency to shock formation in balance

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2.2 Solitary waves

Over one hundred and fifty years ago, while conducting experiments to

determine the most efficient design for canal boats, a young Scottish engineer named John Scott Russell (1808-1882) made a remarkable scientific discovery. Here is an extract from

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Solitary waves

Russell’s report on waves

“I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped - not so the mass of water in the channel which it had put in motion; it

accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity,

assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the

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2.3 Korteweg deVries (KdV) equation

• The wave of translation (or solitary wave) observed by John Scott Russell is described by a nonlinear wave

equation known as the Korteweg-deVries (KdV) equation.

• We review various possible types of nonlinearity in wave equations before studying two specific equations – the KdV and the nonlinear Schrodinger (NLS) equations.

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'wave.dat' 0 ₂₀ 40 ₆₀ 80 100 120 ₁₄₀₀ 0.51 1.5 2 2.53 3.54 -0.20 0.2 0.4 0.6 0.81 1.2 1.4 1.6 1.8

Korteweg deVries (KdV) equation

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Numerical solution (weak dispersive term)

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Effect of nonlinear term u_t = -(1+u)u_x

2 4 6 8 10 -1 -0.5 0.5 1 1.5 u_x(∆t) -(1+u(∆t))u_x(∆t) u(∆t)

The sequence of plots at t = 0, ∆t and 2∆t illustrate how a pulse forms and splits off from the leading edge of a smooth front.

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Korteweg deVries (KdV) equation Effect of dispersive term u_t = - u_xxx

2 4 6 8 10 -1 -0.5 0.5 1 1.5 u(0) -u_xxx(0)

Combined effects of nonlinear and dispersive terms

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Korteweg deVries (KdV) equation Soliton simulations

These simulations come from

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Solution for PBC and sinusoidal initial conditions

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Korteweg deVries (KdV) equation Analytic solution

•KdV equation

•Let the solution be u = u(x,t) and consider a change of variables ξ = x – ct and τ = t

•Call the function in new variables f(ξ,τ)

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• If we convert the change in f brought about by translations through (dx, dt) into changes in f brought about by translations through (dx, dt)

•Since u and f represent the same function the same translation (dx, dt) must produce the same change in either. Hence

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• When transforming the pde from (x, t) to (ξ, τ) we must make the replacements

• In the (x, t) variables a soliton moves along the x axis as time advances

•In the (ξ, τ) variables a soliton is stationary in time provided we choose c in the transformation to be the soliton velocity

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• The conventional form for the KdV equation is • Travelling wave solutions have the form

c is the wave velocity

• Substituting for u in the KdV equation and setting the time derivative to zero we obtain

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• A and B are constants of integration. In order to have a localised traveling wave packet, we impose boundary

conditions: all tend to zero as |ξ| goes to infinity. • To ensure these conditions we set A = B = 0. Solutions also exist at zeros of the polynomial in f.

• The solution with A = B = 0 obeys

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(

)

∫

= +

θ

d c 2 tanh c sech 2 c tanh sech c d becomes c 2f f df 2 2 2

• Make change of variable

θ

2 sech 2 c -f = c x c 2 _o + ± = −

θ

ξ

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(

)

_      − = 2 (x -ct - x_o 2 c sech 2 c f _m

θ

_θ _θ sech -sech tanh

d d e e 2 sech = − = ₋

(

- x _o

)

2 c

_ξ

m = • Rearrange to

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2.4 Nonlinear Schrödinger equation

• The naming of the nonlinear Schrödinger (NLS) equation becomes obvious when it is compared to the time-dependent Schrödinger equation from quantum mechanics

0 ψ V xx ψ 2m t ψ i 0 ψ ψ Q xx ψ P t iψ 2 2 = − + = + + h h

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Nonlinear Schrödinger equation

Derivation from dispersion relation

• Consider the superposition of 2 waves of similar wavenumber and frequency

t]

-cos[kx

t]

-k x

[

cos

2 ψ

ψ

t]

)

(

-k)x

cos[(k

ψ

t]

)

(

-k)x

cos[(k

ψ

2 1 2 1

ω

∆

=

+

∆

−

∆

−

=

∆

+

∆

+

=

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• The NLS is derived from the dispersion relation for the envelope function which has a slow time variation cf the carrier waves

• Suppose that the dispersion relationship is

Make a Taylor expansion of this about k_o and zero intensity

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• Let 2 o ψ , o 2 ψ 2 o , o 2 2 2 o , o g o o ω ω Q ω k ω 2P ω k ω v k – k K ω -ω Ω ∂ ∂ = ∂ ∂ = ∂ ∂ = = = ψ ψ

• Then the Taylor expanded dispersion relation becomes

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• Consider a wavepacket constructed from a small group of waves in slow variables X = εx, T = εt ε <<1

ve carrier wa * d dK -)] T -KX exp[i( ) (K, T) (X, t)] -x exp[i(k * d dK -)] T -KX exp[i( ) (K, d dk -t)] -exp[i(kx ) (k, t) (x, o o Ω ∫ ∞ ∞ Ω Ω = Ω ∫ ∞ ∞ Ω Ω = ∫ ∞ ∞ = ε ε ψ ψ ω ε ε ψ ω ω ω ψ ψ

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• The dispersion relation

2 2 gK PK Q v + +

ψ

= Ω becomes

ψ

ε

ψ

ε

ψ

ε

εψ

ψ

ε

ψ

ε

ψ

ε

ψ

ε

2 2 XX 2 X g T 2 2 2 g Q P v i - i Q X i -P X i v T i + + = +       ∂ ∂ +       ∂ ∂ = ∂ ∂

• Make further change of variables

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ψ

ε

ψ

ε

ψ

ε

εψ

₂ 2 XX 2 X g T - i v P Q i = + + becomes 0 q i 0 Q P i 2 2 = + + = + +

ϕ

τ

ϕ

τ

ϕ

ξξ ξξ

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Nonlinear Schrödinger equation Application to lattice dynamics

[

3

]

1 -n n 3 n 1 n 1 n n 1 n n K(u 2u u ) AK (u u ) (u u ) u m _&& = ₊ − + ₋ − ₊ − − −

• Hooke’s Law plus additional nonlinear term

3 4 2 AKr -Kr dr dU(r) - F(r) r 4 AK Kr 2 1 U(r) − = = + = • Equation of motion

• Solution and dispersion relation

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• We have just seen that introduction of a nonlinear term in the force law for a 1-D chain of atoms leads to a dispersion

relation which depends on |R|2_{. At the website below, use the}

monatomic chain applet to see some of these localised modes. •Intrinsic localised modes in lattice dynamics of crystals

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• Click on monatomic 1-D chains and then on the link in the title to the page (works best with Internet Explorer)

•You will find stationary ILM with

• envelope function (c.f. solutions of NLS equation) is

composed of groups of waves centred on the Brillouin zone boundary (k = π) (group velocity zero)

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• You will also find

• molecular dynamics simulations showing ILM in 3-D crystals (click on 3-D Ionic crystals)

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Application to optical communications

• Read the introductory articles on

• Solitons in optical communications by Ablowitz et al.

• Historical aspects of optical solitons by Hasegawa